In this volume, the authors show that a set of local admissible fields generates a vertex algebra. For an affine Lie algebra \(\tilde{\mathfrak g}\), they construct the corresponding level \(k\) vertex operator algebra and show that level \(k\) highest weight \(\tilde{\mathfrak g}\)-modules are modules for this vertex operator algebra. They determine the set of annihilating fields of level \(k\) standard modules and study the corresponding loop \(\tilde{\mathfrak g}\)-module--the set of relations that defines standard modules. In the case when \(\tilde{\mathfrak g}\) is of type \(A^{(1)}_1\), they construct bases of standard modules parameterized by colored partitions, and as a consequence, obtain a series of Rogers-Ramanujan type combinatorial identities.

Readership

Graduate students and research mathematicians working in representation theory; theoretical physicists interested in conformal field theory.

Table of Contents

• Abstract
• Introduction
• Formal Laurent series and rational functions
• Generating fields
• The vertex operator algebra \(N(k\Lambda_0)\)
• Modules over \(N(k\Lambda_0)\)
• Relations on standard modules
• Colored partitions, leading terms and the main results
• Colored partitions allowing at least two embeddings
• Relations among relations
• Relations among relations for two embeddings
• Linear independence of bases of standard modules
• Some combinatorial identities of Rogers-Ramanujan type
• Bibliography